┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "Treatment_io"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: S, In, Tr, N
[ Info: Parameters: b, nu, d, g, a
[ Info: Inputs: 
[ Info: Outputs: y1, y2
[ Info: Summary of the model:
[ Info: State variables: x1, x2
[ Info: Parameters: a, b, d, c
[ Info: Inputs: 
[ Info: Outputs: y
[ Info: Computing IO-equations
┌ Info: Computed in 14.794947052 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.794947052
[ Info: Computing Wronskians
┌ Info: Computed in 11.624716204 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.624716204
[ Info: Dimensions of the Wronskians [10, 1]
┌ Info: Ranks of the Wronskians computed in 0.030770391 seconds
│   :rank_time = :rank_time
└   rank_times = 0.030770391

⌜ # Computing specializations..  	 Time: 0:00:10[K
✓ # Computing specializations..  	 Time: 0:00:10[K

⌜ # Computing specializations..  	 Time: 0:00:04[K
✓ # Computing specializations..  	 Time: 0:00:04[K
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 1 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 40.
[ Info: Groebner basis computed in 13.239814153 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.473080475 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 4 fractions 4 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.972652471 seconds. Result: true
[ Info: Out of 12 initial generators there are 4 indepdendent
[ Info: The ranking of the new set of generators is 62
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 2 for den.
│ Maximal number of interpolated terms are: 3 for num. and 1 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 5.341926983 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.475743896 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (4, 4)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 4 for num. and 3 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 72.
[ Info: Groebner basis computed in 0.101438923 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003947647 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 8 for num. and 3 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 104.
[ Info: Groebner basis computed in 0.132816762 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004831968 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 21 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (9 in total): Nemo.QQMPolyRingElem[b, nu, d, g, a, S, In, Tr, N]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 9
┌ Info: Final cleaning and simplification of generators. 
└ Out of 20 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 1.60474987 seconds. Result: true
[ Info: Out of 9 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 77
[ Info: The search for identifiable functions concluded in 79.409337429 seconds
[ Info: Processing Treatment_io
┌ Info: Averaging over 1 runs.
│ Using keyword arguments:
│ NamedTuple{(:with_states, :strategy), Tuple{Bool, Tuple{Symbol, Int64}}}
│ (with_states = true, strategy = (:normalforms, 2))
└ ID: (:normalforms, 2)_with_states
[ Info: Computing IO-equations
┌ Info: Computed in 0.00848791 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.00848791
[ Info: Computing Wronskians
┌ Info: Computed in 0.006481304 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.006481304
[ Info: Dimensions of the Wronskians [10, 1]
┌ Info: Ranks of the Wronskians computed in 1.9652e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 1.9652e-5
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 1 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 40.
[ Info: Groebner basis computed in 0.015410113 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001618724 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 4 fractions 4 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.004029379 seconds. Result: true
[ Info: Out of 12 initial generators there are 4 indepdendent
[ Info: The ranking of the new set of generators is 62
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 2 for den.
│ Maximal number of interpolated terms are: 3 for num. and 1 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 0.024949519 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002529538 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (4, 4)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 4 for num. and 3 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 72.
[ Info: Groebner basis computed in 0.066568563 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002509969 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 8 for num. and 3 for den.
│ Maximal number of interpolated terms are: 4 for num. and 1 for den.
└ Points used: 104.
[ Info: Groebner basis computed in 0.114227557 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004404247 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 21 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (9 in total): Nemo.QQMPolyRingElem[b, nu, d, g, a, S, In, Tr, N]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 9
┌ Info: Final cleaning and simplification of generators. 
└ Out of 20 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.010055207 seconds. Result: true
[ Info: Out of 9 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 77
[ Info: The search for identifiable functions concluded in 0.436714014 seconds
┌ Info: Result is
│   result =
│    8-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     N
│     Tr
│     S*g
│     b*S
│     nu + d*g
│     nu*g + nu*a
│     nu + g + a
└     In*g + Tr*g + Tr*a
